0

Monotone Class Definition:

We say that $\mathcal{G}$ is a monotone class if whenever $\{A_{k}\}$ is an increasing and $\{B_{k}\}$ is a decreasing sequence in $\mathcal{G},$ then $\cup A_{k}$ and $\cap B_{k}$ are in $\mathcal{G},$ as well.

My question is:

How can I prove rigorously that this set $$\{(-\infty, a_{n}), (-\infty, a_{n}], \emptyset, [a_{n}, \infty),\mathbb{R}, (a_{n}, \infty)\}, $$

with $(-\infty, a_n)$ such that $(a_n)$ decreasing, $(-\infty, a_n]$ s.t. $(a_n)$ decreasing, $(a_n,\infty) $ s.t. $(a_n)$ increasing, or $[ a_n, \infty)$ s.t. $(a_n)$ increasing.

My trial:

According to intuition, the countable union of increasing unbounded intervals is in my set, it is just $(a_{1}, \infty)$ or $[a_{1}, \infty)$, but still I donot know how to prove it rigorously, so any help in this step will be appreciated.

for the infinite intersection I do not have any intuition and I am unable to prove it rigorously. so any help in this step will also be appreciated.

is a monotone class?

Emptymind
  • 1,901
  • 2
    There is already an answer here: https://math.stackexchange.com/questions/3550749/checking-an-example-of-a-monotone-class/3551059?noredirect=1#comment7317809_3551059 – Kavi Rama Murthy Feb 24 '20 at 23:17
  • @KaviRamaMurthy where is the proof that it is a monotone class there ? I can not see the proof. – Emptymind Feb 24 '20 at 23:41
  • 2
    I'm confused about the definition of this set. Is $(a_n)$ increasing or decreasing (or constant)? Or is there supposed to be an increasing $(a_n)$ and decreasing $(b_n)$, with the set being $${(-\infty, b_n), (-\infty, b_n], \emptyset, [a_n, \infty), (a_n, \infty)},$$ ranging over $n \in \Bbb{N}$? – user744868 Feb 25 '20 at 00:30
  • @KaviRamaMurthy do u have any elaboration for user744868 comment? – Emptymind Feb 25 '20 at 00:32
  • @Emptymind $(a_n, \infty)$ is increasing iff $a_n$ is decreasing; $(-\infty,a_n)$ is decreasing iff $a_n$ is decreasing, and so on. – Kavi Rama Murthy Feb 25 '20 at 05:20
  • and what is the union of the increasing portion and the intersection of the decreasing one? – Emptymind Feb 25 '20 at 05:24

0 Answers0